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Landscape microtemperaments: Difference between revisions

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== Terture ==
== Terture ==
{{See also| Vulture family }}
{{See also| Vulture family }}
Named by [[Xenllium]] in 2021, terture may be described as the {{nowrap| 111 & 159 }} temperament.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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== Slendscape ==
== Slendscape ==
Slendscape tempers out the [[slendroschisma]] (68719476736/68641485507) in addition to landscape comma, and thereby features a period of 1\15.
Named by [[Xenllium]] in 2025, slendscape tempers out the [[slendroschisma]] (68719476736/68641485507) in addition to landscape comma, and thereby features a period of 1\15.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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: mapping generators: ~8575/8192, ~1152/875
: mapping generators: ~8575/8192, ~1152/875


[[Optimal tuning]] ([[CTE]]): ~8575/8192 = 80.0000{{c}}, ~1152/875 = 475.4847{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~8575/8192 = 79.9771{{c}}, ~1152/875 = 475.4832{{c}}
: [[error map]]: {{val| -0.043 -0.022 +0.087 +0.053 }}
* [[CWE]]: ~8575/8192 = 80.0000{{c}}, ~1152/875 = 475.4962{{c}}
: error map: {{val| 0.000 +0.030 +0.175 +0.182 }}


{{Optimal ET sequence|legend=1| 255, 270, 525, 795, 1065 }}
{{Optimal ET sequence|legend=1| 15, 240, 255, 270, 795, 1065, 1335, 2400 }}


[[Badness]] (Smith): 0.058002
[[Badness]] (Sintel): 1.47


=== 11-limit ===
=== 11-limit ===
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Comma list: 3025/3024, 102487/102400, 180224/180075
Comma list: 3025/3024, 102487/102400, 180224/180075


Mapping: {{Mapping| 15 0 17 54 40 | 0 4 3 -2 2 }}
Mapping: {{mapping| 15 0 17 54 40 | 0 4 3 -2 2 }}


Optimal tuning (CTE): ~22/21 = 80.0000{{c}}, ~968/735 = 475.4912{{c}}
Optimal tunings:
* WE: ~22/21 = 79.9991{{c}}, ~968/735 = 475.4915{{c}}
* CWE: ~22/21 = 80.0000{{c}}, ~968/735 = 475.4955{{c}}


{{Optimal ET sequence|legend=0| 255, 270, 525, 795, 1065 }}
{{Optimal ET sequence|legend=0| 15, 240, 255, 270, 795, 1065, 2400e }}


Badness (Smith): 0.026246
Badness (Sintel): 0.868


=== 13-limit ===
=== 13-limit ===
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Comma list: 1716/1715, 3025/3024, 4096/4095, 14641/14625
Comma list: 1716/1715, 3025/3024, 4096/4095, 14641/14625


Mapping: {{Mapping| 15 0 17 54 40 109 | 0 4 3 -2 2 -9 }}
Mapping: {{mapping| 15 0 17 54 40 109 | 0 4 3 -2 2 -9 }}


Optimal tuning (CTE): ~22/21 = 80.0000{{c}}, ~154/117 = 475.4935{{c}}
Optimal tunings:
* WE: ~22/21 = 79.9993{{c}}, ~154/117 = 475.4902{{c}}
* CWE: ~22/21 = 80.0000{{c}}, ~154/117 = 475.4943{{c}}


{{Optimal ET sequence|legend=0| 255, 270, 795, 1065 }}
{{Optimal ET sequence|legend=0| 255, 270, 795, 1065 }}


Badness (Smith): 0.021230
Badness (Sintel): 0.877


== Akjayland ==
== Akjayland ==
{{See also| 21st-octave temperaments }}
{{See also| 21st-octave temperaments }}


Akjayland tempers out the [[akjaysma]] in addition to landscape comma, and thereby features a period of 1\21.  
Named by [[Eliora]] in 2022, akjayland tempers out the [[akjaysma]] in addition to landscape comma, and thereby features a period of 1\21.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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: mapping generators: ~1323/1280, ~131072/91875
: mapping generators: ~1323/1280, ~131072/91875


[[Optimal tuning]] ([[CTE]]): ~1323/1280 = 57.1429{{c}}, ~131072/91875 = 614.9354{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~1323/1280 = 57.1426{{c}}, ~131072/91875 = 614.9336{{c}}
: [[error map]]: {{val| -0.005 -0.012 +0.039 -0.013 }}
* [[CWE]]: ~1323/1280 = 57.1429{{c}}, ~131072/91875 = 614.9360{{c}}
: error map: {{val| 0.000 -0.004 +0.051 +0.002 }}


{{Optimal ET sequence|legend=1| 84, 273, 357, 441, 966, 1407, 1848, 7833, 9681, 11529, 13377c }}
{{Optimal ET sequence|legend=1| 84, 273, 357, 441, 966, 1407, 1848, 7833, 9681, 11529, 13377c }}


[[Badness]] (Smith): 0.0309
[[Badness]] (Sintel): 0.838


=== Vasca ===
=== Vasca ===
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: mapping generators: ~1323/1280, ~6615/5632
: mapping generators: ~1323/1280, ~6615/5632


Optimal tuning (CTE): ~1323/1280 = 57.1429{{c}}, ~6615/5632 = 278.8998{{c}}
Optimal tunings:
* WE: ~1323/1280 = 57.1436{{c}}, ~6615/5632 = 278.9017{{c}}
* CWE: ~1323/1280 = 57.1429{{c}}, ~6615/5632 = 278.8985{{c}}


{{Optimal ET sequence|legend=0| 168, 357, 525, 882, 1407, 2289e }}
{{Optimal ET sequence|legend=0| 168, 357, 525, 882, 1407, 2289e }}


Badness (Smith): 0.0949
Badness (Sintel): 3.14


==== 13-limit ====
==== 13-limit ====
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Mapping: {{mapping| 21 4 39 98 58 107 | 0 6 2 -8 3 -6 }}
Mapping: {{mapping| 21 4 39 98 58 107 | 0 6 2 -8 3 -6 }}


Optimal tuning (CTE): ~336/325 = 57.1429{{c}}, ~168/143 = 278.9058{{c}}
Optimal tunings:
* WE: ~336/325 = 57.1426{{c}}, ~168/143 = 278.9047{{c}}
* CWE: ~336/325 = 57.1429{{c}}, ~168/143 = 278.9060{{c}}


{{Optimal ET sequence|legend=0| 168, 357, 525, 882 }}
{{Optimal ET sequence|legend=0| 168, 357, 525, 882 }}


Badness (Smith): 0.0551
Badness (Sintel): 2.28


==== 17-limit ====
==== 17-limit ====
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Mapping: {{mapping| 21 4 39 98 58 107 120 | 0 6 2 -8 3 -6 -7 }}
Mapping: {{mapping| 21 4 39 98 58 107 120 | 0 6 2 -8 3 -6 -7 }}


Optimal tuning (CTE): ~336/325 = 57.1429{{c}}, ~168/143 = 278.9036{{c}}
Optimal tunings:
* WE: ~336/325 = 57.1429{{c}}, ~168/143 = 278.9037{{c}}
* CWE: ~336/325 = 57.1429{{c}}, ~168/143 = 278.9036{{c}}


{{Optimal ET sequence|legend=0| 168, 357, 525, 882 }}
{{Optimal ET sequence|legend=0| 168, 357, 525, 882 }}


Badness (Smith): 0.0319
Badness (Sintel): 1.62


==== 19-limit ====
==== 19-limit ====
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Mapping: {{mapping| 21 4 39 98 58 107 120 16 | 0 6 2 -8 3 -6 -7 15 }}
Mapping: {{mapping| 21 4 39 98 58 107 120 16 | 0 6 2 -8 3 -6 -7 15 }}


Optimal tuning (CTE): ~336/325 = 57.1429{{c}}, ~168/143 = 278.9036{{c}}
Optimal tunings:
* WE: ~336/325 = 57.1425{{c}}, ~168/143 = 278.8960{{c}}
* CWE: ~336/325 = 57.1429{{c}}, ~168/143 = 278.8976{{c}}


{{Optimal ET sequence|legend=0| 168h, 357, 525, 882, 1407 }}
{{Optimal ET sequence|legend=0| 168h, 357, 525, 882, 1407 }}


Badness (Smith): 0.0270
Badness (Sintel): 1.64


==== 23-limit ====
==== 23-limit ====
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Comma list: 1496/1495, 2376/2375, 2601/2600, 2646/2645, 2926/2925, 3025/3024, 3213/3211
Comma list: 1496/1495, 2376/2375, 2601/2600, 2646/2645, 2926/2925, 3025/3024, 3213/3211


Mapping: {{mapping| 21 4 39 98 58 107 120 16 95 | 0 -6 -2 8 -3 6 7 -15 0 }}
Mapping: {{mapping| 21 4 39 98 58 107 120 16 95 | 0 6 2 -8 3 -6 -7 15 0 }}


Optimal tuning (CTE): ~336/325 = 57.1429{{c}}, ~168/143 = 278.8971{{c}}
Optimal tunings:
* WE: ~336/325 = 57.1422{{c}}, ~168/143 = 278.8949{{c}}
* CWE: ~336/325 = 57.1429{{c}}, ~168/143 = 278.8980{{c}}


{{Optimal ET sequence|legend=0| 168h, 357, 525, 882, 1407 }}
{{Optimal ET sequence|legend=0| 168h, 357, 525, 882, 1407 }}


Badness (Smith): 0.0199
Badness (Sintel): 1.43


== Magnesium ==
== Magnesium ==
: ''For the 5-limit version, see [[12th-octave temperaments#Magnesium (5-limit)]].
: ''For the 5-limit version, see [[12th-octave temperaments #Magnesium (5-limit)]].


Magnesium is named after element 12 for being period 12; however, it is not an extension of the [[atomic]] – the associated comma is {{monzo| -157 -24 84 }} in the 5-limit and the 7 generators together with [[12edo]] major second reach the just perfect fifth, [[3/2]].
Magnesium is named by [[Eliora]] in 2023 after the 12th element for having a 1/12-octave period; however, it is not an extension of the [[atomic]] – the associated comma is {{monzo| -157 -24 84 }} in the 5-limit, and 7 generator steps together with two [[12edo]] semitones reach the [[3/1|3rd]] [[harmonic]]. It may be described as the {{nowrap| 84 & 612 }} temperament, with a [[ploidacot]] signature of dodecaploid gamma-heptacot.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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{{Mapping|legend=1| 12 2 23 58 | 0 7 2 -10 }}
{{Mapping|legend=1| 12 2 23 58 | 0 7 2 -10 }}
: mapping generators: ~138915/131072, 3145728/2734375
: mapping generators: ~138915/131072, ~3145728/2734375


Optimal tuning (CTE): ~138915/131072 = 100.000{{c}}, ~3145728/2734375 = 243.130{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~138915/131072 = 100.0021{{c}}, ~3145728/2734375 = 243.1333{{c}}
: [[error map]]: {{val| +0.025 -0.018 +0.000 -0.039 }}
* [[CWE]]: ~138915/131072 = 100.0000{{c}}, ~3145728/2734375 = 243.1285{{c}}
: error map: {{val| 0.000 -0.055 -0.057 -0.111 }}


{{Optimal ET sequence|legend=1| 84, 360d, 444, 528, 612, 1920, 2532, 10740cd, 13272bcdd, 15804bcdd, 18336bcddd }}
{{Optimal ET sequence|legend=1| 84, 360d, 444, 528, 612, 1920, 2532, 10740cd, 13272bcdd, 15804bcdd, 18336bcddd }}


Badness (Smith): 0.0964
Badness (Sintel): 2.44


== Chromium ==
== Chromium ==
: ''For the 5-limit version, see [[24th-octave temperaments #Chromium (5-limit)]].
: ''For the 5-limit version, see [[24th-octave temperaments #Chromium (5-limit)]].


[[Chromium]] is defined by associating the porcupine comma [[250/243]] to the 24th of an octave. Named after the 24th element for being period 24.
[[Chromium]] is defined by associating the porcupine comma [[250/243]] to the 24th of an octave, and may be described as the {{nowrap| 72 & 624 }} temperament. It was named by [[Eliora]] in 2022 after the 24th element for having a 24th-octave period.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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: mapping generators: ~250/243, ~10/7
: mapping generators: ~250/243, ~10/7


[[Optimal tuning]] ([[CTE]]): ~250/243 = 50.0000{{c}}, ~10/7 = 617.2710{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~250/243 = 49.9992{{c}}, ~10/7 = 617.2714{{c}}
: [[error map]]: {{val| -0.019 -0.142 +0.048 +0.246 }}
* [[CWE]]: ~250/243 = 50.0000{{c}}, ~10/7 = 617.2762{{c}}
: error map: {{val| 0.000 -0.126 +0.067 +0.279 }}


{{Optimal ET sequence|legend=1| 72, …, 480, 552, 624, 1320, 1944d, 3264d }}
{{Optimal ET sequence|legend=1| 72, …, 480, 552, 624, 1320, 1944d, 3264d }}


[[Badness]] (Smith): 0.139
[[Badness]] (Sintel): 3.52


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 9801/9800, 46656/46585, 250047/250000
Comma list: 9801/9800, 46656/46585, 151263/151250


Mapping: {{mapping| 24 1 -6 18 46 | 0 3 5 4 3 }}
Mapping: {{mapping| 24 1 -6 18 46 | 0 3 5 4 3 }}


Optimal tuning (CTE): ~250/243 = 50.0000{{c}}, ~10/7 = 617.2597{{c}}
Optimal tunings:
* WE: ~250/243 = 49.9972{{c}}, ~10/7 = 617.2639{{c}}
* CWE: ~250/243 = 50.0000{{c}}, ~10/7 = 617.2823{{c}}


{{Optimal ET sequence|legend=0| 72, …, 480, 552, 624 }}
{{Optimal ET sequence|legend=0| 72, …, 480, 552, 624 }}


Badness (Smith): 0.0398
Badness (Sintel): 1.32


=== 13-limit ===
=== 13-limit ===
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Mapping: {{mapping| 24 1 -6 18 46 -47 -13 | 0 3 5 4 3 11 }}
Mapping: {{mapping| 24 1 -6 18 46 -47 -13 | 0 3 5 4 3 11 }}


Optimal tuning (CTE): ~250/243 = 50.0000{{c}}, ~10/7 = 617.2869{{c}}
Optimal tunings:
* WE: ~250/243 = 49.9958{{c}}, ~10/7 = 617.2824{{c}}
* CWE: ~250/243 = 50.0000{{c}}, ~10/7 = 617.3161{{c}}


{{Optimal ET sequence|legend=0| 72, …, 480f, 552, 624, 1176de, 1800cdee }}
{{Optimal ET sequence|legend=0| 72, …, 480f, 552, 624, 1176de, 1800cdee }}


Badness (Smith): 0.0293
Badness (Sintel): 1.21


=== 17-limit ===
=== 17-limit ===
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Mapping: {{mapping| 24 1 -6 18 46 -47 -13 | 0 3 5 4 3 11 9 }}
Mapping: {{mapping| 24 1 -6 18 46 -47 -13 | 0 3 5 4 3 11 9 }}


Optimal tuning (CTE): ~35/34 = 50.0000{{c}}, ~10/7 = 617.2732{{c}}
Optimal tunings:
* WE: ~35/34 = 49.9959{{c}}, ~10/7 = 617.2685{{c}}
* CWE: ~35/34 = 50.0000{{c}}, ~10/7 = 617.3015{{c}}


{{Optimal ET sequence|legend=0| 72, …, 480fgg, 552g, 624 }}
{{Optimal ET sequence|legend=0| 72, …, 480fgg, 552g, 624 }}


Badness (Smith): 0.0209
Badness (Sintel): 1.06


== Zinc ==
== Zinc ==
{{See also| 10th-octave temperaments #Neon }}
{{See also| 10th-octave temperaments #Neon | 60th-octave temperaments #Neodymium }}
{{See also| 60th-octave temperaments #Neodymium }}
 
Zinc maybe described as the {{nowrap| 270 & 2190 }} temperament. It was named by [[Eliora]] in 2023 after the 30th element for having a 30th-octave period.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~53747712/52521875 = 40.0000{{c}}, ~216/175 = 364.3896{{c}} (~{{monzo| 21 3 1 -10 }} = 4.3896{{c}})
* [[WE]]: ~53747712/52521875 = 40.0002{{c}}, ~216/175 = 364.3890{{c}} (~{{monzo| 21 3 1 -10 }} = 4.3869{{c}})
* [[CWE]]: ~53747712/52521875 = 40.0000{{c}}, ~216/175 = 364.3879{{c}} (~{{monzo| 21 3 1 -10 }} = 4.3879{{c}})
* [[CWE]]: ~53747712/52521875 = 40.0000{{c}}, ~216/175 = 364.3879{{c}} (~{{monzo| 21 3 1 -10 }} = 4.3879{{c}})


{{Optimal ET sequence|legend=1| 270, 1380, 1650, 1920, 2190, 4650 }}
{{Optimal ET sequence|legend=1| 270, 1380, 1650, 1920, 2190, 4650 }}


[[Badness]] (Smith): 0.0742
[[Badness]] (Sintel): 1.88


=== 11-limit===
=== 11-limit===
Retrieved from "https://en.xen.wiki/w/Landscape_microtemperaments"